Simplifying Negative Exponents: (a^4/b^2)^-8

Question

$(\frac{a^4}{b^2})^{-8}=\text{?}$

00:00 Simplify the following problem

00:03 In order to eliminate a negative exponent

00:07 We'll flip the numerator and the denominator so that the exponent will become positive

00:12 For example

00:16 We'll apply the formula in order to convert from a fraction to a number with a negative exponent

00:21 When there's an exponent on a product of terms, each factor is raised to that power

00:29 We'll apply this formula to our exercise

00:36 When there's a power of a power, the resulting exponent is the product of the exponents

00:44 We'll apply this formula to our exercise and multiply the exponents

00:49 We'll calculate the exponents, and that's the solution to the question

To solve this problem, we'll simplify the expression using exponent rules:

Step 1: Apply the negative exponent rule.
The expression $\left(\frac{a^4}{b^2}\right)^{-8}$ with a negative exponent can be rewritten using the reciprocal:
$\left(\frac{a^4}{b^2}\right)^{-8} = \frac{1}{\left(\frac{a^4}{b^2}\right)^{8}}$
Step 2: Apply the power of a quotient rule.
Using $\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$ , we have:
$\frac{1}{\left(\frac{a^4}{b^2}\right)^{8}} = \frac{1}{\frac{(a^4)^8}{(b^2)^8}} = \frac{1}{\frac{a^{32}}{b^{16}}}$
Step 3: Simplify by taking the reciprocal of the fraction.
Thus, $\frac{1}{\frac{a^{32}}{b^{16}}} = \frac{b^{16}}{a^{32}}$ , which can be rewritten using exponent rules as $b^{16}a^{-32}$ .

Therefore, the simplified expression is $b^{16}a^{-32}$ .

Given the multiple-choice options, the correct choice is the one that matches our final result.

Therefore, the solution to the problem is $b^{16}a^{-32}$ .

$b^{16}a^{-32}$